Transdichotomous Results in Computational Geometry, I: Point Location in Sublogarithmic Time

Abstract

Given a planar subdivision whose coordinates are integers bounded by U ≤ 2 w , we present a linear-space data structure that can answer point location queries in O(min{lg n/ lg lg n, lg U/ lg lg U }) time on the unit-cost RAM with word size w. This is the first result to beat the standard Θ(lg n) bound for infinite precision models.As a consequence, we obtain the first o(n lg n) (randomized) algorithms for many fundamental problems in computational geometry for arbitrary integer input on the word RAM, including: constructing the convex hull of a three-dimensional point set, computing the Voronoi diagram or the Euclidean minimum spanning tree of a planar point set, triangulating a polygon with holes, and finding intersections among a set of line segments. Higher-dimensional extensions and applications are also discussed.Though computational geometry with bounded precision input has been investigated for a long time, improvements have been limited largely to problems of an orthogonal flavor. Our results surpass this long-standing limitation, answering, for example, a question of Willard (SODA'92).

Persistent Predecessor Search and Orthogonal Point Location on the Word RAM

“…Note that the query bound is much better than Chan and Pǎtraşcu's bound O(min{ log U/ log log U , log n/ log log n}) for general planar point location [20]. The triangle fatness assumption is common in the literature.…”

“…On the word RAM, Chan and Pǎtraşcu [20] were the first to obtain sublogarithmic query time O(min{ log U/ log log U , log n/ log log n}) with linear space for arbitrary, nonorthogonal planar subdivisions with coordinates from {1, . .…”

“…For example, for orthogonal 2-d point location, the trivial method of dividing into O(n) vertical slabs enables queries to be answered by a search in x followed by a search in y, using total space O(n 2 ); for persistent predecessor search, the trivial method of building a new data structure after every update has O(n) update time. We can then apply known techniques to lower space (or update time in the persistent problem), for example, via sampling, separators, or exponential search trees, as described by Chan and Pǎtraşcu [20]. With a constant number of rounds of reduction, the space usage can be brought down to O(n 1+ε ).…”

“…This step is standard (e.g., in 1-d van Emde Boas trees, it is analogous to the final transformation from "x-fast" to "y-fast tries" [61]). Chan and Pǎtraşcu [20] have described three general approaches to lowering space for planar point location: random sampling, planar graph separators, and an approach based on 1-d exponential search trees. De Berg et al [16] have specifically employed the graph separator approach for orthogonal planar point location.…”

“…An interesting direction is to explore offline (or batched ) query problems, in the spirit of [19,21]. The version of 1-d persistent predecessor search where the updates and queries are all given in advance is equivalent to the version of orthogonal 2-d point location where n query points are given in advance.…”